To solve for the missing frequency [tex]\( X \)[/tex], we need to use the information given about the average daily wage and the other known wages and number of workers.
### Step-by-Step Solution:
1. Identify the known quantities:
- Average daily wage: [tex]\( ₹41 \)[/tex]
- Daily wages: [tex]\( ₹20, ₹30, ₹40, ₹50, ₹60, ₹70 \)[/tex]
- Number of workers for each wage: [tex]\( 8, 12, X, 10, 6, 4 \)[/tex]
2. Calculate the total number of workers:
[tex]\[
\text{Total number of workers} = 8 + 12 + X + 10 + 6 + 4 = 40 + X
\][/tex]
3. Calculate the total wage sum using the average wage formula:
The average wage is given by the formula:
[tex]\[
\text{Average wage} = \frac{\text{Total wage sum}}{\text{Total number of workers}}
\][/tex]
Therefore:
[tex]\[
₹41 = \frac{\text{Total wage sum}}{40 + X}
\][/tex]
So the total wage sum is:
[tex]\[
41 (40 + X) = 1640 + 41X
\][/tex]
4. Set up the equation for the total wage sum:
The total wages for the known workers can be calculated by multiplying the number of workers by their corresponding wages:
[tex]\[
\text{Total wage sum} = (8 \times 20) + (12 \times 30) + (X \times 40) + (10 \times 50) + (6 \times 60) + (4 \times 70)
\][/tex]
Breaking this down:
[tex]\[
\text{Total wage sum} = 160 + 360 + 40X + 500 + 360 + 280
\][/tex]
Combining the constants:
[tex]\[
\text{Total wage sum} = 1660 + 40X
\][/tex]
5. Equate the total wage sums to solve for [tex]\( X \)[/tex]:
Set the total wage sum calculated from the average wage equation equal to the total wage sum equation:
[tex]\[
1640 + 41X = 1660 + 40X
\][/tex]
Solve for [tex]\( X \)[/tex]:
[tex]\[
1640 + 41X = 1660 + 40X
\][/tex]
Subtract [tex]\( 40X \)[/tex] from both sides:
[tex]\[
1640 + X = 1660
\][/tex]
Subtract 1640 from both sides:
[tex]\[
X = 20
\][/tex]
### Conclusion:
The missing frequency [tex]\( X \)[/tex] is [tex]\( 20 \)[/tex].
